The Complexity of Power Up, Pt. 2 – A ‘Simple’ Example

Yesterday, I wrote about how No-Limit Hold’em may be incredibly difficult to solve in a strict sense, but can be made a manageable task by reducing it to a set of simpler games. The branching of the game tree can be kept under control by limiting the player to a few discrete bet sizes, and its height can be reduced by taking things on a street by street basis and employing heuristics like equity and range-balancing to set short-term goals.

I went on to assert that this is not possible for PokerStars’s new game, Power Up. Even though it plays short-stacked and has fast blind levels, it can’t be attacked by the same approaches used for something like Spin & Go, because the interactions of the powers both force us to consider more options at each decision point, and the need to budget energy prevents us from taking things street-by-street.

The most solvable scenario

We can see the problem very clearly if we take one of the simplest situations in conventional Hold’em, and think about how we’d go about solving it in Power Up. It is, in fact, the first scenario arising in actual play to which players applied software techniques to find an equilibrium solution: The short-stacked heads-up push-or-fold decision.

So, we have one player on the button (which is also the small blind in heads-up play), and the other in the big blind. Effective stacks are 6 big blinds. We assume, based on experience, that limping or raising less than all-in is not a good strategy for the button. Therefore, the button should either go all-in or fold. If he folds, the big blind has no decision to make. If he shoves, then she also has a binary decision, to call or fold.

This limited number of ways that the hand can play out mean that we only need to calculate two ranges to “solve” the situation. We need to know what hands the button should shove with, and which hands the big blind should call with. Game theory tells us that we should find a pair of ranges that are in equilibrium with one another, such that the weakest hands included in each are just slightly better than break-even against the other range, and the strongest hands being folded are just slightly lossy. Once we’ve found those ranges, neither player can do any better by deviating from their strategy.

The traditional approach

There are several ways to go about finding a solution, but the easiest (though not the fastest or most accurate) with computer assistance is to start from a very simplistic strategy and iterate from there.

First, assume that the big blind calls with 100% of hands. Figure out what range of hands is profitable for the button to shove with under that assumption. Now, go back and look at the big blind’s hands, and figure out which calls are bad (loss-making) with that presumed range for the button. Remove those hands to get big blind’s new range. Now, look at the button’s range again and see which hands he’s shoving which are no longer profitable, and whether there are hands he was folding which would now be profitable. Keep doing this until you reach the point where neither player has cause to update their range any further: That’s your equilibrium.

Do this for various stack sizes and you’ll end up with something like the various charts you’ll find on poker training sites, showing which hands are correct for the button to shove for a given stack size, and which ones are correct for the big blind to call with.

No simple scenarios in Power Up

Given the number of powers available and the considerations which arise when players have enough energy to use multiple powers, a full examination of a 6 big blind heads-up situation in Power Up is too complicated even to write out let alone solve. Let me make things considerably simpler, then, and show that even a stripped-down Power Up is still incredibly complex.

Let’s assume that both players have 5 energy, and pretend that their powers are face up: Both have an EMP and a Reload. The former costs 3 energy and blocks further power use on the current street, and the latter costs 5 energy and allows a player to redraw one or both hole cards.

There are now several things that the button might want to do. Let’s take a look at the options and what sorts of hands he might want to take each line with.

Fold: It’s probably safe to assume that it’s not an effective use of energy by the button to use Reload to swap both cards when we have a bad hand. There is also no need to be balanced when folding, since it leaves the opponent no decision. Therefore, we can assume that there is still a folding range in Power Up, consisting of hands too weak to do anything else.

All-In: Sometimes we may wish to go all-in directly without using a power first. The primary example would be pocket Aces. With very strong hands, we don’t much mind if the opponent gets to Reload, and we don’t want to discourage a call and waste energy by using EMP. If we don’t balance our range, however, our opponent will simply fold to this play, so as well as our premium hands, we need some bluffs. Some Ace-blocker hands may make sense, to reduce the chances that our opponent’s Reload improves her, and maybe a small suited connector or two, as these are in better-than-average shape when our opponent does wake up with (or Reloads into) a monster.

EMP + All-In: Most of the time that we’re going to be going all-in, we’ll want to protect ourselves with an EMP. Since our opponent is stuck with the cards she was dealt, most hands that are profitable in conventional Hold’em will be profitable here. Thus, this range should consist more or less of those hands we would shove in conventional Hold’em, minus those that end up in one of the other ranges.

Limp: There’s little advantage to limping when this short-stacked in conventional Hold’em, but it serves a purpose in Power Up, namely to retain the option to use Reload later. Since our opponent might either check back or go all-in, we want to include some hands which might benefit from a Reload after seeing the flop – offsuit connectors spring to mind – and others which would like to Reload before calling a shove – King-rag combinations for instance. However, she might attempt to thwart our plan by using her own EMP before going all-in, so we need to be trapping with some strong hands for balance. Pocket Jacks, Queens and Kings seem like a good idea, as they’re strong enough that we don’t want to waste an EMP only to get a fold, but compared to pocket Aces, they fare worse against an opponent who Reloads one card and then calls.

You can see already that this is much more complicated than conventional Hold’em, because we’re dividing our hands into four ranges, rather than just two. But now, consider the four situations these ranges present to our opponent, and what she can do in response:

If we fold, the opponent still has no choices to make.

If we go all-in, the opponent can call, fold, or Reload (and then call or fold).

If we EMP + all-in, the opponent can only call or fold.

If we limp, the opponent can check, go all-in, EMP + all-in, or Reload (and then either check or go all-in).

Thus, a complete strategy for our opponent must contain all the following ranges:

Fold to all-in

Call all-in

Reload to all-in

Fold to all-in after Reload

Call all-in after Reload

Fold to EMP + all-in

Call EMP all-in

Check to limp

All-in to limp

EMP + all-in to limp

Reload to limp

Check to limp after Reload

All-in to limp after Reload

And, of course, if we limp and the opponent goes all-in, we can call, fold, Reload-call or Reload-fold, and each of these will depend on whether our opponent only went all-in, or first used an EMP or a Reload. That means we, as the button, have another ten ranges to establish (not twelve, since we can’t Reload if she uses EMP).

In total, then, assuming we found a good approach to generate an equilibrium for this simplified game, displaying the resulting strategy would require: for the button, four charts containing a total of 14 ranges; plus for the big blind, six charts containing a total of 13 ranges.

Not just hard to memorize, but hard to compute

That sounds bad enough, but we’re still assuming we can actually calculate the equilibria in the first place, and that is going to be much harder than it was for conventional Hold’em.

We can no longer take a simple iterative approach, as most of these tables contain more than two ranges; if a hand is not profitable in its current range, we don’t necessarily know which of the other ranges to move it into. We could choose the most profitable play to make with it, but this can easily lead to rock-paper-scissors scenarios where one player’s best move with hand A depends on what the opponent does with hand B and vice versa. In these situations, it’s not guaranteed that we’ll find an equilibrium without resorting to mixed strategies, which requires more sophisticated approaches.

There’s a second complication in that there are now certain lines (limp-check and limp-Reload-check) which result in post-flop play. We can’t as easily determine a value for these situations as we can for those where either both players end up all-in or someone folds. Thus, before we can even begin to solve the preflop situation, we have to look at how things will typically go down postflop, either through even more complicated theoretical approaches, or perhaps empirically, though this would require a huge volume of hand histories and is likely to give results which change over time as play style evolves.

Finally, there’s the matter of energy, and what it’s worth. Shoving and getting a fold is worth more than EMP-shoving and getting a fold, for instance, because although the chips won are the same, winning the pot without an EMP saves three energy. Computing a solution will require establishing how much that energy is worth in future chip-EV. Again here, we could try a theoretical approach or an empirical one, but neither is easy.

Keeping in mind that this is actually a massively simplified and unrealistic Power Up scenario (in that in real play, players won’t know what powers the other holds), it should be clear that Power Up is a monstrously more difficult problem to solve than Hold’em.

A return to the good old days of human intuition

Given the power of modern AI, it’s likely that very strong Power Up-playing machines would be possible, just as they are for conventional Hold’em. However, the resulting strategies are so complex that they will not be memorisable by humans the way conventional hand range charts are.

Human Power Up players will therefore always be reliant on heuristics (that is, “rules of thumb”) and intuition to play well, just as they were in the old days of Hold’em. It’s not just that GTO solver software hasn’t yet been developed for Power Up; rather, it’s that it never will exist, at least not the way it does for conventional short-stacked formats. Players will make software tools to help them get better, naturally, that much is inevitable. But the output those tools generate will, at best, help top players formulate good heuristics; it will never come down to a set of tables laying out what to do in every situation.

Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Dartmouth, Nova Scotia, Canada.

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